Cubic critical, narrow range

Average Error: 28.6 → 0.3

Time: 8.2s

Precision: binary64

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\frac{-c}{b + \sqrt{{b}^{2} - \sqrt[3]{27 \cdot \left(\left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)}}} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/
  (- c)
  (+
   b
   (sqrt (- (pow b 2.0) (cbrt (* 27.0 (* (* c a) (* (* c a) (* c a))))))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return -c / (b + sqrt(pow(b, 2.0) - cbrt(27.0 * ((c * a) * ((c * a) * (c * a))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.6

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Simplified 28.6

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

3. Applied flip--_binary64 28.6

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \cdot \frac{0.3333333333333333}{a} \]

4. Applied associate-*l/_binary64 28.6

   \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \]

5. Simplified 0.6

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, 0\right) \cdot \frac{0.3333333333333333}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

6. Taylor expanded in a around 0 0.3

   \[\leadsto \frac{\color{blue}{-1 \cdot c}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

7. Simplified 0.3

   \[\leadsto \frac{\color{blue}{-c}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

8. Taylor expanded in a around 0 0.3

   \[\leadsto \frac{-c}{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(c \cdot a\right)}} + b} \]

9. Applied add-cbrt-cube_binary64 0.3

   \[\leadsto \frac{-c}{\sqrt{{b}^{2} - 3 \cdot \color{blue}{\sqrt[3]{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}} + b} \]

10. Applied add-cbrt-cube_binary64 0.3

    \[\leadsto \frac{-c}{\sqrt{{b}^{2} - \color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}} \cdot \sqrt[3]{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}} + b} \]

11. Applied cbrt-unprod_binary64 0.3

    \[\leadsto \frac{-c}{\sqrt{{b}^{2} - \color{blue}{\sqrt[3]{\left(\left(3 \cdot 3\right) \cdot 3\right) \cdot \left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)\right)}}} + b} \]

12. Final simplification 0.3

    \[\leadsto \frac{-c}{b + \sqrt{{b}^{2} - \sqrt[3]{27 \cdot \left(\left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)}}} \]

Reproduce

herbie shell --seed 2022095 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))